(Solution) - a Find the entropy of a set of N oscillators -(2025 Original AI-Free Solution)

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(a) Find the entropy of a set of N oscillators of frequency ? as a function of the total quantum number n. Use the multiplicity function (1.55) and make the Stirling approximation log N! ? N log N ? N. Replace N ? 1 by N.
(b) Let U denote the total energy nh? of the oscillators. Express the entropy as ?(U, N). Show that the total energy at temperature ? is
U = Nh? / ex(h?/?)?1 (42)
This is the Plank result; it is derived again in Chapter 4 by a powerful method that does not require us to find the multiplicity function.